Test 3 - Faculty.lasierra.edu

(a) What sample size would he need to estimate the percentage within ±3% with 95% confidence?(assume he doesn’t have an initial estimate for the true proportion)(b) Same question as (a), but within ±1% with 95% confidence.(c) Same question as (a), but within ±3% with 99% confidence.(d) Repeat questions (a) through (c), but assume to start with that a news poll estimated that 27%of Americans support sending additional troups, and that the president used this as a good startingestimate for p.(e) Approximately what sample size do you think the Gallup Organization uses to create polls thatare accurate to ±3 percent 19 times out of 20?Answer. (a) For this part, z c = 1.96, the desired E = .03, and there is no initial estimate for p, son = 1 ( zc) ( ) 2 21 1.96= n = ≈ 1067.14 E 4 .03This means he should use a sample size of 1068.(b) In this case E = .01, and son = 1 4( zc) 2 1 = n =E 4( ) 2 1.96= 9604..01Notice the huge increase in sample size from (a) to 9604 to get within 1 percent.(c) Same as (a), except z c = 2.58 and so n = 1849 using the same formula.( zc) 2(d) Now use the formula n = pq where we use .27 as an estimate for p, and .73 as an estimateEfor q. To to get an accurace of .03 with 95 percent confidence, then( zc) ( ) 2 2 1.96n = pq ≈ (.27)(.73) ≈ 841.3E.03and so a sample size of n = 842 would be sufficient. Use the same formula for the other parts, butwith E = .01 in the next calculation, and z c = 2.58 in the final calculation.(e) They will never need a random sample larger than 1068 (see (a)). If there is a good estimate forp, they may be able to use a smaller sample (see (d)). If you look at the fine print of some polls,you will see that often a sample size of approximately 1000 is used.III.D.3. (a) What sample size would be needed to estimate the mean from a population withstandard deviation of 100 with a maximum error of 5 with 95% confidence?(b) What sample size from the same population would be needed to estimate the mean with amaximum error of 10 with 95% confidence?(c) In general, what effect would increasing a sample size by a factor of 16 have on the maximumerror E?(d) In general, how much must a sample size be increased to cut the maximum error by one-half?Answer. (a) We use the formula n =n = 1537.( zc σE) ( ) 2, 2 1.96 · 100and so n = = 1536.64 and so we use5